Yes, it is correct to ask "What is your mother tongue?" However, I agree with you, it does sound a bit strange. That will depend a bit on which English speaker you are talking to though (where they are from). It is correct though.
Other ways you can ask the same question are:
- What is your native language?
- What is your first language? (Personally, I think this one sounds best.)
- What language did you speak growing up? (Only ask this if they clearly have multiple languages.)